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Math 2280-002, Spring 2013 Euler's Method The goal of this worksheet is to demonstate how MAPLE can be used to implement Euler's method. In particular, we'll consider the IVP , We can solve this initial value problem explicitly by hand using integrating factors, but that would be tedious since we would have to integrate by parts multiple times. Instead, let's have MAPLE do the work for us. > (1) > That means our IVP has the particular solution Now let's use Euler's method to approximate this solution and then check how close our approximation is. Euler's method on the interval [a,b] for the IVP , with step size is as follows. The step size is The algorithm is where > > > is the slope of the tangent line at > Now that we have initialized these variables, use a for-loop to run the Euler algorithm. > x1=1.1 y1=1.2 x2=1.2 y2=1.46641 x3=1.3 y3=1.82041 x4=1.4 y4=2.28806 x5=1.5 y5=2.90103 x6=1.6 y6=3.69738 x7=1.7 y7=4.72248 x8=1.8 y8=6.02994 x9=1.9 y9=7.68269 x10=2 y10=9.75417 > Graph the solution and the approximation. > > > 10 8 6 4 2 1 2 x Let's compare the value we get from Euler's method with the actual value of the solution at . > (2) > (3) > (4) > (5) > As you can see, this isn't a particularly good approximation even when N=10. How can you modify the code above for the improved Euler's method? What about for Runge-Kutta?